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cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Historia</span> </button> <ul id="toc-Historia-sublist" class="vector-toc-list"> <li id="toc-Problema_Isoperimétrico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problema_Isoperimétrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Problema Isoperimétrico</span> </div> </a> <ul id="toc-Problema_Isoperimétrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Braquistócrona" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Braquistócrona"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Braquistócrona</span> </div> </a> <ul id="toc-Braquistócrona-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formulación_xeneral" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulación_xeneral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formulación xeneral</span> </div> </a> <button aria-controls="toc-Formulación_xeneral-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Formulación xeneral</span> </button> <ul id="toc-Formulación_xeneral-sublist" class="vector-toc-list"> <li id="toc-Espacios_funcionales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_funcionales"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Espacios funcionales</span> </div> </a> <ul id="toc-Espacios_funcionales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estremos_relativos_débiles_y_fuertes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estremos_relativos_débiles_y_fuertes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Estremos relativos débiles y fuertes</span> </div> </a> <ul id="toc-Estremos_relativos_débiles_y_fuertes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ver tamién</span> </div> </a> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referencies</span> </div> </a> <button aria-controls="toc-Referencies-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Referencies</span> </button> <ul id="toc-Referencies-sublist" class="vector-toc-list"> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaces_esternos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enllaces_esternos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Enllaces esternos</span> </div> </a> <ul id="toc-Enllaces_esternos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteníu" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cálculu de variaciones</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 45 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-45" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">45 llingües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AA%D8%BA%D9%8A%D8%B1%D8%A7%D8%AA" title="حساب المتغيرات – árabe" lang="ar" hreflang="ar" data-title="حساب المتغيرات" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%B0%D0%BB%D1%8B_%D0%B8%D2%AB%D3%99%D0%BF%D0%BB%D3%99%D0%BC%D3%99" title="Вариациалы иҫәпләмә – bashkir" lang="ba" hreflang="ba" data-title="Вариациалы иҫәпләмә" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D1%8B%D1%8F%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D0%B5_%D0%B7%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D0%BD%D0%B5" title="Варыяцыйнае злічэнне – bielorrusu" lang="be" hreflang="be" data-title="Варыяцыйнае злічэнне" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE_%D1%81%D0%BC%D1%8F%D1%82%D0%B0%D0%BD%D0%B5" title="Вариационно смятане – búlgaru" lang="bg" hreflang="bg" data-title="Вариационно смятане" data-language-autonym="Български" data-language-local-name="búlgaru" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/C%C3%A0lcul_de_variacions" title="Càlcul de variacions – catalán" lang="ca" hreflang="ca" data-title="Càlcul de variacions" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Varia%C4%8Dn%C3%AD_po%C4%8Det" title="Variační počet – checu" lang="cs" hreflang="cs" data-title="Variační počet" data-language-autonym="Čeština" data-language-local-name="checu" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BB%D0%BB%D0%B5_%D1%88%D1%83%D1%82%D0%BB%D0%B0%D0%B2" title="Вариацилле шутлав – chuvash" lang="cv" hreflang="cv" data-title="Вариацилле шутлав" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Variationsrechnung" title="Variationsrechnung – alemán" lang="de" hreflang="de" data-title="Variationsrechnung" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CF%84%CF%89%CE%BD_%CE%BC%CE%B5%CF%84%CE%B1%CE%B2%CE%BF%CE%BB%CF%8E%CE%BD" title="Λογισμός των μεταβολών – griegu" lang="el" hreflang="el" data-title="Λογισμός των μεταβολών" data-language-autonym="Ελληνικά" data-language-local-name="griegu" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Calculus_of_variations" title="Calculus of variations – inglés" lang="en" hreflang="en" data-title="Calculus of variations" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Variada_kalkulo" title="Variada kalkulo – esperanto" lang="eo" hreflang="eo" data-title="Variada kalkulo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/C%C3%A1lculo_de_variaciones" title="Cálculo de variaciones – español" lang="es" hreflang="es" data-title="Cálculo de variaciones" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Variatsioonarvutus" title="Variatsioonarvutus – estoniu" lang="et" hreflang="et" data-title="Variatsioonarvutus" data-language-autonym="Eesti" data-language-local-name="estoniu" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bariazioen_kalkulu" title="Bariazioen kalkulu – vascu" lang="eu" hreflang="eu" data-title="Bariazioen kalkulu" data-language-autonym="Euskara" data-language-local-name="vascu" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1%D8%A7%D8%AA" title="حساب تغییرات – persa" lang="fa" hreflang="fa" data-title="حساب تغییرات" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Variaatiolaskenta" title="Variaatiolaskenta – finlandés" lang="fi" hreflang="fi" data-title="Variaatiolaskenta" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Calcul_des_variations" title="Calcul des variations – francés" lang="fr" hreflang="fr" data-title="Calcul des variations" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/C%C3%A1lculo_de_variaci%C3%B3ns" title="Cálculo de variacións – gallegu" lang="gl" hreflang="gl" data-title="Cálculo de variacións" data-language-autonym="Galego" data-language-local-name="gallegu" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%95%D7%A8%D7%99%D7%90%D7%A6%D7%99%D7%95%D7%AA" title="חשבון וריאציות – hebréu" lang="he" hreflang="he" data-title="חשבון וריאציות" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%9A%E0%A4%B0%E0%A4%A3-%E0%A4%95%E0%A4%B2%E0%A4%A8" title="विचरण-कलन – hindi" lang="hi" hreflang="hi" data-title="विचरण-कलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vari%C3%A1ci%C3%B3sz%C3%A1m%C3%ADt%C3%A1s" title="Variációszámítás – húngaru" lang="hu" hreflang="hu" data-title="Variációszámítás" data-language-autonym="Magyar" data-language-local-name="húngaru" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Calcolo_delle_variazioni" title="Calcolo delle variazioni – italianu" lang="it" hreflang="it" data-title="Calcolo delle variazioni" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%89%E5%88%86%E6%B3%95" title="変分法 – xaponés" lang="ja" hreflang="ja" data-title="変分法" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D0%B5%D1%81%D0%B5%D0%BF%D1%82%D0%B5%D1%83" title="Вариациялық есептеу – kazaquistanín" lang="kk" hreflang="kk" data-title="Вариациялық есептеу" data-language-autonym="Қазақша" data-language-local-name="kazaquistanín" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%80%EB%B6%84%EB%B2%95" title="변분법 – coreanu" lang="ko" hreflang="ko" data-title="변분법" data-language-autonym="한국어" data-language-local-name="coreanu" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Kalkulu_tal-varjazzjonijiet" title="Kalkulu tal-varjazzjonijiet – maltés" lang="mt" hreflang="mt" data-title="Kalkulu tal-varjazzjonijiet" data-language-autonym="Malti" data-language-local-name="maltés" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Variatierekening" title="Variatierekening – neerlandés" lang="nl" hreflang="nl" data-title="Variatierekening" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Variasjonsrekning" title="Variasjonsrekning – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Variasjonsrekning" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Variasjonsregning" title="Variasjonsregning – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Variasjonsregning" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_wariacyjny" title="Rachunek wariacyjny – polacu" lang="pl" hreflang="pl" data-title="Rachunek wariacyjny" data-language-autonym="Polski" data-language-local-name="polacu" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/C%C3%A0lcol_dle_variassion" title="Càlcol dle variassion – piamontés" lang="pms" hreflang="pms" data-title="Càlcol dle variassion" data-language-autonym="Piemontèis" data-language-local-name="piamontés" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%A1lculo_variacional" title="Cálculo variacional – portugués" lang="pt" hreflang="pt" data-title="Cálculo variacional" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Calcul_varia%C8%9Bional" title="Calcul variațional – rumanu" lang="ro" hreflang="ro" data-title="Calcul variațional" data-language-autonym="Română" data-language-local-name="rumanu" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE%D0%B5_%D0%B8%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Вариационное исчисление – rusu" lang="ru" hreflang="ru" data-title="Вариационное исчисление" data-language-autonym="Русский" data-language-local-name="rusu" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Calculus_of_variations" title="Calculus of variations – Simple English" lang="en-simple" hreflang="en-simple" data-title="Calculus of variations" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Varia%C4%8Dn%C3%BD_po%C4%8Det" title="Variačný počet – eslovacu" lang="sk" hreflang="sk" data-title="Variačný počet" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Variacijski_ra%C4%8Dun" title="Variacijski račun – eslovenu" lang="sl" hreflang="sl" data-title="Variacijski račun" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Analiza_e_variacionit" title="Analiza e variacionit – albanu" lang="sq" hreflang="sq" data-title="Analiza e variacionit" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Varijacijski_ra%C4%8Dun" title="Varijacijski račun – serbiu" lang="sr" hreflang="sr" data-title="Varijacijski račun" data-language-autonym="Српски / srpski" data-language-local-name="serbiu" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Variationskalkyl" title="Variationskalkyl – suecu" lang="sv" hreflang="sv" data-title="Variationskalkyl" data-language-autonym="Svenska" data-language-local-name="suecu" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D1%96%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Варіаційне числення – ucraín" lang="uk" hreflang="uk" data-title="Варіаційне числення" data-language-autonym="Українська" data-language-local-name="ucraín" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_t%C3%ADnh_bi%E1%BA%BFn_ph%C3%A2n" title="Phép tính biến phân – vietnamín" lang="vi" hreflang="vi" data-title="Phép tính biến phân" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamín" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8F%98%E5%88%86%E6%B3%95" title="变分法 – chinu wu" lang="wuu" hreflang="wuu" data-title="变分法" data-language-autonym="吴语" data-language-local-name="chinu wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%98%E5%88%86%E6%B3%95" title="变分法 – chinu" lang="zh" hreflang="zh" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/16px-Robot_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/24px-Robot_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/32px-Robot_icon.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ast" dir="ltr"><p>El <b>cálculu de variaciones</b> o <b>cálculu variacional</b> ye un <a href="/wiki/Problema_matem%C3%A1ticu" title="Problema matemáticu">problema matemáticu</a> consistente en buscar <a href="/w/index.php?title=Estremos_d%27una_funci%C3%B3n_m%C3%A1ximos_y_m%C3%ADnimos&action=edit&redlink=1" class="new" title="Estremos d'una función máximos y mínimos (la páxina nun esiste)">Estremos d'una función máximos y mínimos</a> (o más xeneralmente estremos relativos) de <a href="/w/index.php?title=Funcional_(matem%C3%A1tica)&action=edit&redlink=1" class="new" title="Funcional (matemática) (la páxina nun esiste)">funcionales</a> continuos definíos sobre dalgún <a href="/w/index.php?title=Espaciu_funcional&action=edit&redlink=1" class="new" title="Espaciu funcional (la páxina nun esiste)">espaciu funcional</a>. Constitúin una xeneralización del cálculu elemental de máximos y mínimos de <a href="/w/index.php?title=Funci%C3%B3n_real&action=edit&redlink=1" class="new" title="Función real (la páxina nun esiste)">funciones reales</a> d'una variable </p><p><br /> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=1" title="Editar seición: Historia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=1" title="Editar el código fuente de la sección: Historia"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El cálculu de variaciones desenvolver a partir del problema de la <a href="/w/index.php?title=Curva_braquist%C3%B3crona&action=edit&redlink=1" class="new" title="Curva braquistócrona (la páxina nun esiste)">curva braquistócrona</a>, plantegáu primeramente por <a href="/w/index.php?title=Johann_Bernoulli&action=edit&redlink=1" class="new" title="Johann Bernoulli (la páxina nun esiste)">Johann Bernoulli</a> (1696). Darréu esti problema captó l'atención de <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jakob Bernoulli</a> y el <a href="/w/index.php?title=Guillaume_de_l%27H%C3%B4pital&action=edit&redlink=1" class="new" title="Guillaume de l'Hôpital (la páxina nun esiste)">Marqués de L'Hôpital</a>, anque foi <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> el primeru qu'ellaboró una teoría del cálculu variacional. Les contribuciones de Euler empecipiar en 1733 cola so <i>Elementa Calculi Variationum</i> ('Elementos del cálculu de variaciones') que da nome a la disciplina. </p><p><a href="/w/index.php?title=Joseph_Louis_Lagrange&action=edit&redlink=1" class="new" title="Joseph Louis Lagrange (la páxina nun esiste)">Lagrange</a> contribuyó estensamente a la teoría y <a href="/w/index.php?title=Adrien-Marie_Legendre&action=edit&redlink=1" class="new" title="Adrien-Marie Legendre (la páxina nun esiste)">Legendre</a> (1786) asitió un métodu, non dafechu satisfactoriu pa estremar ente máximos y mínimos. <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> y <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibniz</a> tamién emprestaron atención a esti asuntu.<sup id="cite_ref-brunt_1-0" class="reference"><a href="#cite_note-brunt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Otros trabayos destacaos fueron los de <a href="/w/index.php?title=Vincenzo_Brunacci&action=edit&redlink=1" class="new" title="Vincenzo Brunacci (la páxina nun esiste)">Vincenzo Brunacci</a> (1810), <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1829), <a href="/w/index.php?title=Sim%C3%A9on_Poisson&action=edit&redlink=1" class="new" title="Siméon Poisson (la páxina nun esiste)">Siméon Poisson</a> (1831), <a href="/w/index.php?title=Mija%C3%ADl_Ostrogradski&action=edit&redlink=1" class="new" title="Mijaíl Ostrogradski (la páxina nun esiste)">Mijaíl Ostrogradski</a> (1834) y <a href="/wiki/Carl_Gustav_Jakob_Jacobi" title="Carl Gustav Jakob Jacobi">Carl Jacobi</a> (1837). Un trabayu xeneral particularmente importante ye'l de <a href="/w/index.php?title=Sarrus&action=edit&redlink=1" class="new" title="Sarrus (la páxina nun esiste)">Sarrus</a> (1842) que foi resumíu por <a href="/w/index.php?title=Cauchy&action=edit&redlink=1" class="new" title="Cauchy (la páxina nun esiste)">Cauchy</a> (1844). Otros trabayos destacaos posteriores son los de <a href="/w/index.php?title=Strauch&action=edit&redlink=1" class="new" title="Strauch (la páxina nun esiste)">Strauch</a> (1849), <a href="/w/index.php?title=Jellett&action=edit&redlink=1" class="new" title="Jellett (la páxina nun esiste)">Jellett</a> (1850), <a href="/w/index.php?title=Otto_Hesse&action=edit&redlink=1" class="new" title="Otto Hesse (la páxina nun esiste)">Otto Hesse</a> (1857), <a href="/w/index.php?title=Alfred_Clebsch&action=edit&redlink=1" class="new" title="Alfred Clebsch (la páxina nun esiste)">Alfred Clebsch</a> (1858) y <a href="/w/index.php?title=Carll&action=edit&redlink=1" class="new" title="Carll (la páxina nun esiste)">Carll</a> (1885), anque quiciabes el más importante de los trabayos mientres el sieglu XIX ye'l de <a href="/w/index.php?title=Weierstrass&action=edit&redlink=1" class="new" title="Weierstrass (la páxina nun esiste)">Weierstrass</a>. Esti importante trabayu foi una referencia estándar y ye'l primeru que trata'l cálculu de variaciones sobre una base firme y rigoroso. Los <a href="/wiki/Problemes_de_Hilbert" title="Problemes de Hilbert">problema 20 y 23 de Hilbert</a> plantegaos en 1900 aguiyaron dellos desarrollos posteriores.<sup id="cite_ref-brunt_1-1" class="reference"><a href="#cite_note-brunt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Mientres el sieglu XX, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>, <a href="/w/index.php?title=Leonida_Tonelli&action=edit&redlink=1" class="new" title="Leonida Tonelli (la páxina nun esiste)">Leonida Tonelli</a>, <a href="/w/index.php?title=Henri_Lebesgue&action=edit&redlink=1" class="new" title="Henri Lebesgue (la páxina nun esiste)">Henri Lebesgue</a> y <a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a>, ente otros, fixeron contribuciones notables.<sup id="cite_ref-brunt_1-2" class="reference"><a href="#cite_note-brunt-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="/w/index.php?title=Marston_Morse&action=edit&redlink=1" class="new" title="Marston Morse (la páxina nun esiste)">Marston Morse</a> aplicó'l cálculu de variaciones a lo qu'anguaño se conoz como <a href="/w/index.php?title=Teor%C3%ADa_de_Morse&action=edit&redlink=1" class="new" title="Teoría de Morse (la páxina nun esiste)">teoría de Morse</a>.<sup id="cite_ref-ferguson_2-0" class="reference"><a href="#cite_note-ferguson-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/w/index.php?title=Lev_Semenovich_Pontryagin&action=edit&redlink=1" class="new" title="Lev Semenovich Pontryagin (la páxina nun esiste)">Lev Semenovich Pontryagin</a>, <a href="/w/index.php?title=R._Tyrrell_Rockafellar&action=edit&redlink=1" class="new" title="R. Tyrrell Rockafellar (la páxina nun esiste)">Ralph Rockafellar</a> y Clarke desenvolvieron nueves ferramientes matemátiques dientro de la teoría del control óptimo, xeneralizando'l cálculu de variaciones.<sup id="cite_ref-ferguson_2-1" class="reference"><a href="#cite_note-ferguson-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Problema_Isoperimétrico"><span id="Problema_Isoperim.C3.A9trico"></span>Problema Isoperimétrico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=2" title="Editar seición: Problema Isoperimétrico" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=2" title="Editar el código fuente de la sección: Problema Isoperimétrico"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r4219085">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=Isoperimetr%C3%ADa&action=edit&redlink=1" class="new" title="Isoperimetría (la páxina nun esiste)">Isoperimetría</a></div> <p>¿Cuál ye l'área máxima <i>A</i> que puede arrodiase con una curva de llargor <i>L</i> dada? Si nun esisten restricciones adicionales, la solución ye: </p> <style data-mw-deduplicate="TemplateStyles:r4219090">.mw-parser-output .ecuacion{padding:5px 10px;background-color:var(--background-color-base);color:var(--color-base);margin-left:30px;margin-bottom:0.8em;margin-top:0.5em;min-width:50%}.mw-parser-output .ecuacion .referencia{float:right;width:10%;text-align:end}.mw-parser-output .ecuacion cite{font-style:normal}</style><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {L^{2}}{4\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {L^{2}}{4\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c23d4e806cda78da19ca27890bd4228de6f29d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.315ex; height:5.676ex;" alt="{\displaystyle A={\frac {L^{2}}{4\pi }}}"></span> </p> </blockquote> <p>Que ye'l valor que se llogra pa un círculu de radiu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=L/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=L/2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55261c78426504109e223e19e431fc5253030e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.102ex; height:2.843ex;" alt="{\displaystyle R=L/2\pi }"></span>. </p><p>Si impónense restricciones adicionales la solución ye distinta. Un exemplu ye si suponemos que <i>L</i> considérase sobre una función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> y los estremos de les curva tán sobre los puntos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a,0),B=(b,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a,0),B=(b,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71274de17dbb4f75571e809604cafd96bdfd8c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.977ex; height:2.843ex;" alt="{\displaystyle A=(a,0),B=(b,0)}"></span> onde la distancia ente ellos ta dada. Ye dicir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=L\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369c425dcbc66cf656bc9b3da5a06da0c7ac4c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.576ex; height:2.176ex;" alt="{\displaystyle AB=L\,}"></span>. El problema de topar una curva que maximice l'área ente ella y la exa x sería, topar una función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> de cuenta que: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mi>I</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0edbd94617f1172835c275c10031436e1282bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.12ex; height:6.509ex;" alt="{\displaystyle \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}"></span> </p> </blockquote> <p>coles restricciones: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{llargor d'arcu}}\\f(a)=f(b)=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>llargor d'arcu</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{llargor d'arcu}}\\f(a)=f(b)=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8418ba4af45c6b8a45d2a5cc9db4fd2c0ff9ddeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.238ex; height:6.509ex;" alt="{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{llargor d'arcu}}\\f(a)=f(b)=0\end{cases}}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Braquistócrona"><span id="Braquist.C3.B3crona"></span>Braquistócrona</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=3" title="Editar seición: Braquistócrona" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=3" title="Editar el código fuente de la sección: Braquistócrona"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div><p> El problema de la curva <a href="/w/index.php?title=Braquist%C3%B3crona&action=edit&redlink=1" class="new" title="Braquistócrona (la páxina nun esiste)">braquistócrona</a> remontar a <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">J. Bernoulli</a> (1696). Referir a atopar una curva nel planu cartesianu que vaya del puntu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29b6070ac1b7ca23c822ac47b0b1bf4593ab526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.265ex; height:2.843ex;" alt="{\displaystyle P=(x_{0},y_{0})}"></span> al orixe de cuenta que un puntu material que s'esmuz ensin resfregón sobre ella tarda'l menor tiempu posible en dir de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> al orixe. Usando principios de <a href="/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica">mecánica clásica</a> el problema puede formulase como, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"></p><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mi>T</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msqrt> <mn>2</mn> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b1d04f1f08b5584f4af926fb849bb2030ff36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.721ex; height:7.509ex;" alt="{\displaystyle \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}"></span> </p> </blockquote> <p>onde <i>g</i> ye la gravedá y les restricciones son, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d308c32c9894b88115262081194321ae7d9bbf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{0})=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1661f200e31065bfb7291ec00a2e829cfe18f97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.764ex; height:2.843ex;" alt="{\displaystyle f(x_{0})=y_{0}}"></span>. Hai que notar qu'en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> esiste una <a href="/w/index.php?title=Singularid%C3%A1_matem%C3%A1tica&action=edit&redlink=1" class="new" title="Singularidá matemática (la páxina nun esiste)">singularidá</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Formulación_xeneral"><span id="Formulaci.C3.B3n_xeneral"></span>Formulación xeneral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=4" title="Editar seición: Formulación xeneral" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=4" title="Editar el código fuente de la sección: Formulación xeneral"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unu de los problemes típicos en <a href="/wiki/C%C3%A1lculu_diferencial" title="Cálculu diferencial">cálculu diferencial</a> ye'l d'atopar el valor de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> pal cual la función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f(x)\,}"></span> algama un valor estremu (máximu o mínimu). Nel cálculu de variaciones el problema ye atopar una <a href="/wiki/Funci%C3%B3n_(matem%C3%A1tiques)" class="mw-redirect" title="Función (matemátiques)">función</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> pa la cual un <a href="/w/index.php?title=Funcional&action=edit&redlink=1" class="new" title="Funcional (la páxina nun esiste)">funcional</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J[f]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J[f]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ea0bf8b1b1e72b887230a17788d39eb3ba7fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle J[f]\,}"></span> algame un valor estremu. El funcional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J[f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J[f]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de95b8fa7148a0649ae7783e6d6d0fb67ab5e18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle J[f]}"></span> ta compuestu por una integral que depende de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, de la función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> y dalgunes de les sos derivaes. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="referencia">(<cite id="Equation_1a"><a href="#Eqnref_1a">1a</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mi>I</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80931d48f42382bd8b4dc9f25a16bf0d9337241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.074ex; height:6.509ex;" alt="{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}"></span> </p> </blockquote> <p>Onde la función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> pertenez a dalgún espaciu de funciones (<a href="/w/index.php?title=Espaciu_de_Banach&action=edit&redlink=1" class="new" title="Espaciu de Banach (la páxina nun esiste)">espaciu de Banach</a>, <a href="/wiki/Espaciu_de_Hilbert" title="Espaciu de Hilbert">espaciu de Hilbert</a>), y tanto ella como les sos derivaes pueden tener restricciones. Esta fórmula integral pue ser más complicada dexando a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ser un vector, y polo tanto incluyendo derivaes parciales pa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="referencia">(<cite id="Equation_1b"><a href="#Eqnref_1b">1b</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mi>J</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7653ed90cdff2c9c5c416f6600426cfb095d0451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.729ex; height:6.176ex;" alt="{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Espacios_funcionales">Espacios funcionales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=5" title="Editar seición: Espacios funcionales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=5" title="Editar el código fuente de la sección: Espacios funcionales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La fundamentación rigorosa del cálculu de variaciones rique considerar variedaes diferenciales lliniales de <a href="/w/index.php?title=Dimensi%C3%B3n_infinita&action=edit&redlink=1" class="new" title="Dimensión infinita (la páxina nun esiste)">dimensión infinita</a>. De fechu el puntu de partida del cálculu de variaciones ye un teorema d'analís funcional que prueba que ye posible considerar una curva nun espaciu funcional (e.g. trayeutoria nel <a href="/w/index.php?title=Espaciu_f%C3%A1sico&action=edit&redlink=1" class="new" title="Espaciu fásico (la páxina nun esiste)">espaciu fásico</a>) a cencielles como una función con una variable adicional, concretamente:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <style data-mw-deduplicate="TemplateStyles:r4219309">.mw-parser-output .teorema-contenedor{min-width:50%;max-width:77%}.mw-parser-output .teorema{padding:.5em 2em .5em 1.5em;padding-right:2em;padding-left:1.5em;padding-bottom:0.5em;padding-top:0.5em;border:1px solid var(--border-color-base,#49768C);font-family:Georgia,serif}.mw-parser-output .teorema-pie{margin-top:-1em;text-align:right}</style> </p> <table class="teorema-contenedor"> <tbody><tr> <td><blockquote class="teorema"> <p>La categoría formada por <a href="/w/index.php?title=Espaciu_vectorial_conveniente&action=edit&redlink=1" class="new" title="Espaciu vectorial conveniente (la páxina nun esiste)">espacios vectoriales convenientes</a> y <a href="/w/index.php?title=Funci%C3%B3n_nidia&action=edit&redlink=1" class="new" title="Función nidia (la páxina nun esiste)">funciones nidies</a> ente ellos ye cerrada pol productu cartesianu, de tal manera que se tien la siguiente biyección natural:<i></i> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(Y\times F,G)\approx C^{\infty }(Y,C^{\infty }(F,G))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>×</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(Y\times F,G)\approx C^{\infty }(Y,C^{\infty }(F,G))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55862721e0ac112e327ef3bc2ebfeb7068470095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.17ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(Y\times F,G)\approx C^{\infty }(Y,C^{\infty }(F,G))}"></span></dd></dl> </blockquote> <p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle Y,F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>Y</mi> <mo>,</mo> <mi>F</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle Y,F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff194eeb0a458ec2f73543aa1e9aaefec4eaca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.942ex; height:1.843ex;" alt="{\displaystyle \scriptstyle Y,F}"></span> son espacios vectoriales convenientes y la biyección anterior ye un difeomorfismo. </p> </blockquote> </td></tr></tbody></table> <p>El teorema anterior puede aplicase por casu al <a href="/w/index.php?title=Principiu_de_m%C3%ADnima_aici%C3%B3n&action=edit&redlink=1" class="new" title="Principiu de mínima aición (la páxina nun esiste)">principiu de mínima aición</a> onde trata d'atopase la trayeutoria posible nel espaciu de fases que fai mínima la integral d'aición. Dicha trayeutoria ye una curva nidia nel espaciu de trayectories <i>Y</i>, considerando agora: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>F</mi> <mo>=</mo> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e4c9d34b4251ef056da5885b4a0a09ddd888e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.762ex; height:2.843ex;" alt="{\displaystyle Y=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }"></span> </p> </blockquote> <p>Tiense que'l problema de minimización puede amenorgase a embrivir una cierta función real <i>f</i> de variable real: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mi>δ</mi> <mi>q</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="2em" /> <mi>S</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">[</mo> <mi>q</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849760f0cfd3f0afa338b070d6fad2f405858011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:84.67ex; height:6.509ex;" alt="{\displaystyle f_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Estremos_relativos_débiles_y_fuertes"><span id="Estremos_relativos_d.C3.A9biles_y_fuertes"></span>Estremos relativos débiles y fuertes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=6" title="Editar seición: Estremos relativos débiles y fuertes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=6" title="Editar el código fuente de la sección: Estremos relativos débiles y fuertes"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un problema variacional rique que'l funcional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle J(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle J(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc5774343635f21b135989a7293f6cfa59f4a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.224ex; height:2.176ex;" alt="{\displaystyle \scriptstyle J(f)}"></span> tea definíu sobre un <a href="/w/index.php?title=Espaciu_de_Banach&action=edit&redlink=1" class="new" title="Espaciu de Banach (la páxina nun esiste)">espaciu de Banach</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (V,\|\cdot \|_{V})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (V,\|\cdot \|_{V})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0acea78c75183c72778ae5939a475906d20bb93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.292ex; height:2.343ex;" alt="{\displaystyle \scriptstyle (V,\|\cdot \|_{V})}"></span> fayadizu. La <a href="/w/index.php?title=Norma_vectorial&action=edit&redlink=1" class="new" title="Norma vectorial (la páxina nun esiste)">norma vectorial</a> de dichu espaciu ye lo que dexa definir rigorosamente si una solución ye un mínimu o un máximu relativu. Por casu una función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4885ad35ba3bb0582fffe651fd23170c13179d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.637ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f_{0}}"></span> ye un <b>mínimu relativu</b> si esiste un ciertu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7307c8f50fe1fbd19076bd7b3292a93b81046e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.842ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \delta >0}"></span> tal que, pa toa función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>f</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb782e08eca6897b057d996121da1dbbc94a6f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.904ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f}"></span> cumplir que: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>δ</mi> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e87b4239f5987cdb93b183f7b7ea2f10d62aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.176ex; height:2.843ex;" alt="{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=7" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=7" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Charles_Augustin_de_Coulomb&action=edit&redlink=1" class="new" title="Charles Augustin de Coulomb (la páxina nun esiste)">Charles Augustin de Coulomb</a></li> <li><a href="/w/index.php?title=Ecuaciones_de_Euler-Lagrange&action=edit&redlink=1" class="new" title="Ecuaciones de Euler-Lagrange (la páxina nun esiste)">Ecuaciones de Euler-Lagrange</a></li> <li><a href="/w/index.php?title=Derivada_funcional&action=edit&redlink=1" class="new" title="Derivada funcional (la páxina nun esiste)">Derivada funcional</a></li> <li><a href="/wiki/Mec%C3%A1nica_de_suelos" title="Mecánica de suelos">Mecánica de suelos</a></li> <li><a href="/w/index.php?title=Teor%C3%ADa_de_Mohr-Coulomb&action=edit&redlink=1" class="new" title="Teoría de Mohr-Coulomb (la páxina nun esiste)">Teoría de Mohr-Coulomb</a></li> <li><a href="/w/index.php?title=Torsi%C3%B3n_mec%C3%A1nica&action=edit&redlink=1" class="new" title="Torsión mecánica (la páxina nun esiste)">Torsión mecánica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencies">Referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=8" title="Editar seición: Referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=8" title="Editar el código fuente de la sección: Referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-brunt-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-brunt_1-0">1,0</a></sup> <sup><a href="#cite_ref-brunt_1-1">1,1</a></sup> <sup><a href="#cite_ref-brunt_1-2">1,2</a></sup></span> <span class="reference-text"><cite style="font-style:normal">van Brunt, Bruce (2004). <i>The Calculus of Variations</i>. <a href="/w/index.php?title=Springer_Science%2BBusiness_Media&action=edit&redlink=1" class="new" title="Springer Science+Business Media (la páxina nun esiste)">Springer</a>. <a href="/wiki/Especial:FuentesDeLibros/0-387-40247-0" title="Especial:FuentesDeLibros/0-387-40247-0">ISBN 0-387-40247-0</a>.</cite></span> </li> <li id="cite_note-ferguson-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ferguson_2-0">2,0</a></sup> <sup><a href="#cite_ref-ferguson_2-1">2,1</a></sup></span> <span class="reference-text"><a href="/w/index.php?title=Plant%C3%ADa:Cita_arXiv&action=edit&redlink=1" class="new" title="Plantía:Cita arXiv (la páxina nun esiste)">Plantía:Cita arXiv</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">A. Kriegl y P. Michor, 1989, p. 3</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=9" title="Editar seición: Bibliografía" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&action=edit&section=9" title="Editar el código fuente de la sección: Bibliografía"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A. Kriegl y P. W. Michor: <a rel="nofollow" class="external text" href="http://www.mat.univie.ac.at/~michor/aspects.pdf">"Aspects of the theory of inifinite dimensional manifolds"</a>, <i>Differential Geometry and its Applications</i>, <b>1</b>, 1991, páxs. 159-176.</li> <li><a href="/w/index.php?title=Leonida_Tonelli&action=edit&redlink=1" class="new" title="Leonida Tonelli (la páxina nun esiste)">Leonida Tonelli</a>: <a rel="nofollow" class="external text" href="http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath&idno=ACQ6956">Fondamenti di calcolo delle variazioni</a>, N. Zanichelli, 1921-23</li> <li>Todhunter, I. <a rel="nofollow" class="external text" href="http://www.archive.org/details/histroyofthecalc033379mbp">A history of the calculus of variations</a>, Chelsea, 1861</li> <li>Carll, L. B. <a rel="nofollow" class="external text" href="http://www.archive.org/details/treatiseonthecal032865mbp">A Treatise On The Calculus Of Variations</a> John Wiley & sons, 1881</li> <li>Hancock, H. <a rel="nofollow" class="external text" href="http://www.archive.org/details/151181775">Lectures on the calculus of variations (the Weierstrassian theory)</a> Cincinnati University Press, 1904</li> <li>Bolza, O <a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/ACM2513.0001.001">Lectures on the calculus of variations</a>, Chicago University Press, 1904</li> <li>Byerly, W. Y. <a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/ACQ6938.0001.001">Introduction to the calculus of variations</a> <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, 1917</li> <li>Weinstock, R. <a rel="nofollow" class="external text" href="http://www.archive.org/details/calculusofvariat033563mbp">Calculus Of Variations With Applications To Physics And Engineering</a>, McGrawHill, 1952</li> <li>Hadamard J. y Fréchet, M. <a rel="nofollow" class="external text" href="http://www.archive.org/details/leconssurlecalcu00hadarich">Leçons sur le calcul des variations</a> (francese) Hermann, 1910</li> <li>Fomin, S.V. and Gelfand, I.M.: Calculus of Variations, Dover Publ., 2000</li> <li>Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1 – 98</li> <li>Charles Fox: An Introduction to the Calculus of Variations, Dover Publ., 1987</li> <li><a href="/w/index.php?title=Giuseppe_Buttazzo&action=edit&redlink=1" class="new" title="Giuseppe Buttazzo (la páxina nun esiste)">Giuseppe Buttazzo</a>, <a href="/w/index.php?title=Gianni_Dal_Maso&action=edit&redlink=1" class="new" title="Gianni Dal Maso (la páxina nun esiste)">Gianni Dal Maso</a>, <a href="/w/index.php?title=Ennio_De_Giorgi&action=edit&redlink=1" class="new" title="Ennio De Giorgi (la páxina nun esiste)">Ennio De Giorgi</a>. <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/calcolo-delle-variazioni_(8Enciclopedia-Novecento)/">Variazioni, calcolo delle</a>, <i>Enciclopedia del Novecento</i>, II Supplemento (1998), <a href="/w/index.php?title=Istituto_dell%27Enciclopedia_italiana_Treccani&action=edit&redlink=1" class="new" title="Istituto dell'Enciclopedia italiana Treccani (la páxina nun esiste)">Istituto dell'Enciclopedia italiana Treccani</a></li> <li><a href="/w/index.php?title=Gianni_Dal_Maso&action=edit&redlink=1" class="new" title="Gianni Dal Maso (la páxina nun esiste)">Gianni Dal Maso</a>, <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/calcolo-delle-variazioni_%28Enciclopedia-della-Scienza-y-della-Tecnica%29/">Variazioni, calcolo delle</a>, <i>Enciclopedia della Scienza y della Tecnica</i>, (2007), <a href="/w/index.php?title=Istituto_dell%27Enciclopedia_italiana_Treccani&action=edit&redlink=1" class="new" title="Istituto dell'Enciclopedia italiana Treccani (la páxina nun esiste)">Istituto dell'Enciclopedia italiana Treccani</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Enllaces_esternos">Enllaces esternos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculu_de_variaciones&veaction=edit&section=10" title="Editar seición: Enllaces esternos" class="mw-editsection-visualeditor"><span>editar</span></a><span 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title="wikidata:Q216861">Q216861</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Calculus_of_variations">Calculus of variations</a></span></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Library_of_Congress_Control_Number" class="mw-redirect" title="Library of Congress Control Number">LCCN</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85018809">sh85018809</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00563089">00563089</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Library_of_the_Czech_Republic" class="mw-redirect" title="National Library of the Czech Republic">NKC</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph126994">ph126994</a></span></li> <li><b>Diccionarios y enciclopedies</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclopedia_Brit%C3%A1nica" class="mw-redirect" title="Enciclopedia Británica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/calculus-of-variations-mathematics">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q216861" class="extiw" title="wikidata:Q216861">Q216861</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Calculus_of_variations">Calculus of variations</a></span></span></li></ul> </div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Sacáu de «<a dir="ltr" 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